Solve the following problem in spherical coordinates, by separation of variables and choose the appropriate values of the constants to obtain a physical solution. Show that the equation for θ can reproduce the Legendre equation and the associated Legendre equation. ∇^2φ(r, θ, φ) = 0.
Using the generating function of the Legendre polynomial, obtain a general expression for Pn(0).
Prove all parts of equation 5.8 on page 570 of the book.
Find a general expression for the following integrals:
∫[-1]^1 xPl(x)Pn(x)dx
∫[-1]^1 x^2Pl(x)Pn(x)dx
The Dirac delta function is defined as: δ(x - a) =
0 if x ≠a
∞ if x = a
∫[x1]^x2 f(x)δ(x - a)dx = f(a) if a ∈ (x1, x2)
Write δ(x - a) as a Fourier series.